On Donating Regions: Lagrangian Flux through a Fixed Curve

Abstract

Lagrangian flux through a fixed curve segment naturally gives rise to the notion of a donating region, a compact point set in which each particle will pass through the curve and contribute to the flux. A precise geometric determination of the donating region has not been available until now. The author proposes an explicit, constructive, and analytical definition of the donating region based on two characteristic curves of the flow field, viz. streaklines and timelines. It is also shown that, within a time interval of finite size, each particle in the proposed donating region has a net effect of going across the fixed curve once. The presented analysis might be potentially useful for flow topology studies, Lagrangian flux calculation, and explicit interface tracking.

On generalized donating regions: Classifying Lagrangian fluxing particles through a fixed curve in the plane

“…This intuitive argument is the only justification of (4) in the original paper [13]. A later work by the author [11] numerically verifies (4) for incompressible flows.…”

“…1 for three examples of fluxing indices. In the original paper [13,Def. 4.2], the dot product of the normal vector of LN and the velocity vector of the fluxing particle is used in determining the fluxing indices; this definition does not apply if the normal vector of LN does not exist, or, the velocity vector of the particle at a proper intersection aligns with the tangent vector of LN at the same intersection.…”

“…4.2], the dot product of the normal vector of LN and the velocity vector of the fluxing particle is used in determining the fluxing indices; this definition does not apply if the normal vector of LN does not exist, or, the velocity vector of the particle at a proper intersection aligns with the tangent vector of LN at the same intersection. For these cases, Definition 1.2 still holds, and hence it is more general than the original definition in [13].…”

“…For the case that all Lagrangian particles have their fluxing indices |n| ≤ 1, the problem of finding the flux sets F n LN (t 0 , k) has been solved in [13], where a curvilinear polygon D LN called the donating region (DR) was defined from the characteristic curves of the ODE (1) and shown to be equivalent to F ±1 LN (t 0 , k). The DR theory has been applied to analyzing a family of interface tracking methods [12] in multiphase-flow simulations.…”

“…(See [13].) A fluxing particle to a fixed simple curve LN over the time interval (t 0 , t 0 +k) is a particle p whose pathline Φ +k t 0 (p) properly intersects LN at least once.…”